## Patrick Honner’s Pi day exercise in 4d

[This is a quick post I wrote while my younger son was at a little math enrichment activity. Sorry it looks like it was written in a hurry and not proof read . . . ]

Earlier today my older son and I played around with Patrick Honner’s Pi Day exercise:

That project is here:

Patrick Honner’s Pi Day Exercise

After we finished my son wondered about extending the exercise to 4 dimensions!

But, extending to 4 dimensions isn’t as easy as it seems. For one thing, the “volume” and “surface area” of a 4 dimensional sphere involve $\latex \pi^2$ not $\pi$:

“Volume” = $(1/2) \pi^2 R^4$

“Surface Area” = $2 \pi^2 R^3$

So, we’ll modify Honner’s 3d $\pi$ formula to be $\pi^2$ = (1/128) (Surface Area^4) / (Volume^3). That’ll give us a value for $\pi^2$ and then we can compute $\pi$.

So, I found the “volume” and “surface area” of the 4 dimensional regular Polytopes here:

Polytopes

Calculating $"\pi"$ for the regular 4 dimensional polytopes gave values of approximately:

5-Cell: 8.63

8-Cell: 5.66

16-Cell: 4.62

24-Cell: 4.00

120-Cell: 3.38

600-Cell: 3.24

We’ve actually made a 3D version of the 120-cell with our Zometool set:

That project is here, and maybe helps see that the shape is getting sort of spherical.

A Stellated 120-Cell made from our Zometool set

Another way to see some of these 4-dimensional shapes is to check out the game Hypernom:

Using Hypernom to get kids talking about math

Anyway, thanks for Patrick Honner for a fun Pi day!